Pattern-equivariant (PE) cohomology is a well-established tool with which to
interpret the Čech cohomology groups of a tiling space in a highly geometric way.
We consider homology groups of PE infinite chains and establish Poincaré duality
between the PE cohomology and PE homology. The Penrose kite and dart
tilings are taken as our central running example; we show how through this
formalism one may give highly approachable geometric descriptions of the
generators of the Čech cohomology of their tiling space. These invariants are also
considered in the context of rotational symmetry. Poincaré duality fails over
integer coefficients for the “ePE homology groups” based upon chains which
are PE with respect to orientation-preserving Euclidean motions between
patches. As a result we construct a new invariant, which is of relevance to
the cohomology of rotational tiling spaces. We present an efficient method
of computation of the PE and ePE (co)homology groups for hierarchical
tilings.
